Laboratories
A laboratory on this site is a figure with the population known, so the error is visible. The controls change one piece of a model. Where a formula exists, the figure uses it. A simulation or a normal approximation is labelled as one.
The kernel density page came first. The other three are the same idea for a prior, a transition matrix, and a confidence interval.
Kernel density
A Gaussian kernel turns a sample into a density. The bandwidth h is the choice. A small h leaves a bump on the observations: high variance, low bias. A large h washes the modes out. The page keeps the sample still and moves h, and it plots that sample’s integrated squared error against the mean integrated squared error of a sample of the same size. The estimate does not use the true density. Both errors do.
Beta–Binomial updating
A coin has unknown heads probability \theta, and the prior is a Beta distribution. After x heads in n tosses the posterior is \mathrm{Beta}(\alpha+x,\beta+n-x). The posterior mean is a weighted average of the prior mean and x/n, with weight (\alpha+\beta)/(\alpha+\beta+n) on the prior. Future tosses are predicted by the beta–binomial average over that posterior, which is wider than a binomial that fixes \theta at the posterior mean.
A Markov chain
On a finite set of states the whole process is a transition matrix P. From a chosen start, the distribution after t steps is the corresponding row of P^{t}. A stationary distribution satisfies \pi=\pi P. When the chain is one closed aperiodic class, every start forgets itself and P^{t} settles on \pi. A periodic chain can have a stationary distribution while P^{t} keeps oscillating. Two closed classes mean two stationary distributions, and one path’s time average follows the class the path happens to enter.
Coverage of a confidence interval
The confidence is the probability, before the sample is drawn, that the random interval contains the parameter. This page draws the interval repeatedly from a known population and marks the misses. For a normal sample the t interval hits with probability exactly equal to the stated level. Using a normal quantile with the sample standard deviation s hits less often. For a proportion, the exact coverage of the Wald and Wilson intervals is a sum over the binomial distribution, and Wald’s sum falls away from the stated level when the probability is near 0 or 1.